Lemma 3.1 ([1]}, Theorem 5.10) 
Let \(\lbrace X_i\rbrace _{i\in [n]}\)  be the number of balls in bins \(i\in [n]\)  when \(m\)  balls are independently and uniformly placed into \(n\)  bins. Let \(\lbrace Y_i\rbrace _{i\in [n]}\)  be independent \(\mathrm {Poisson}(m/n)\)  random variables. For any monotone set \(S\subseteq [n]\) , we have
\({\mathbb {P}}((X_1, \cdots , X_n)\in S)\le 2{\mathbb {P}}((Y_1, \cdots , Y_n)\in S).\) 

